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Stochastic Integration with Respect to Fractional Brownian Motion

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Stochastic Integration with Respect to Fractional Brownian Motion STOCHASTIC INTEGRATION WITH RESPECT TO FRACTIONAL BROWNIAN MOTION PHILIPPE CARMONA, LAURE COUTIN, AND GERARD´ MONTSENY ABSTRACT. For every value of the Hurst index H (0;1) we define a stochastic integral with respect to fractional Brownian2 motion of index H. We do so by approximating fractional Brownian motion by semi- martingales. Then, for H > 1=6, we establish an Ito’sˆ change of variables formula, which is more precise than Privault’s Ito formula [24] (established for every H > 0), since it only involves anticipating integrals with respect to a driving Brownian motion. RESUM´ E´ Pour tout H (0;1), nous construisons une integrale´ stochas- tique par rapport au mouvement2 Brownien fractionnaire de parametre` de Hurst H. Cette integrale´ est basee´ sur l’approximation du mouvement Brownien fractionnaire par une suite de semi-martingales. Ensuite, pour H > 1=6, nous etablissons´ une formule d’Itoˆ qui precise´ celle obtenue par Privault [24], au sens ou` elle ne comporte que des integrales´ anticipantes par rapport a` un mouvement Brownien directeur. 1. INTRODUCTION Fractional Brownian motion was originally defined and studied by Kol- mogorov [14] within a Hilbert space framework. Fractional Brownian mo- tion of Hurst index H (0;1) is a centered Gaussian process W H with co- variance 2 1 E W HW H = (t2H + s2H t s 2H)(s;t 0) t s 2 − j − j ≥ 1 (for H = 2 we obtain standard Brownian motion). Fractional Brownian motion has stationary increments E (W H(t) W H(s))2 = t s 2H (s;t 0); − j − j ≥ and is H-self similar 1 ( W H(ct);t 0)=(d W H(t);t 0)(for all c > 0): cH ≥ ≥ Date: December 6, 2001. 1991 Mathematics Subject Classification. Primary 60G15, 60H07, 60H05 ; Secondary 60J65, 60F25 . Key words and phrases. Gaussian processes, Stochastic Integrals, Malliavin Calculus, fractional integration. 1 2 P. CARMONA, L. COUTIN, AND G. MONTSENY The Hurst parameter H accounts not only for the sign of the correlation of the increments, but also for the regularity of the sample paths. Indeed, 1 1 for H > 2 , the increments are positively correlated, and for H < 2 they are negatively correlated. Furthermore, for every β (0;H), its sample paths are almost surely Holder¨ continuous with index 2β. Finally, it is worthy of 1 note that for H > 2 , according to Beran’s definition [3], it is a long memory 2H 2 process: the covariance of increments at distance u decrease as u − . These significant properties make fractional Brownian motion a natural can- didate as a model of noise in mathematical finance (see Comte and Re- nault [5], Rogers [26]), and in communication networks (see, for instance, Leland, Taqqu and Willinger [16]). Recently, there has been numerous attempts at defining a stochastic integral 1 H with respect to fractional Brownian motion. Indeed, for H = 2 W is not a semi-martingale (see, e.g., example 2 of section 4.9.13 of Liptser6 and Shyr- iaev [19]), and usual Ito’sˆ stochastic calculus may not be applied. However, the integral t H (1.1) Z a(s)dW (s) 0 may be defined for suitable a. On the one hand, since W H has almost its sample paths Holder¨ continuous of index β, for any β < H, the integral (1.1) exists in the Riemann-Stieljes sense (path by path) if almost every sample 1 β path of a has finite p-variation with p + > 1 (see Young [32]): this is the 1 approach used by Dai and Heyde [8] and Lin [18] when H > 2 . Let us recall that the p-variation of a function f over an interval [0;t] is the least upper p bound of sums ∑i f (xi) f (xi 1) over all partitions 0 = x0 < x1 < ::: < j − − j xn = T. A recent survey of the important properties of Riemann-Stieltjes in- tegral is the concentrated advanced course of Dudley and Norvaisa [11]. An extension of Riemann-Stieltjes integral has been defined by Zahle¨ [33], see also the recent work of Ruzmaikina [29], by means of composition formu- las, integration by parts formula, Weyl derivative formula concerning frac- tional integration/differentiation, and the generalized quadratic variation of Russo and Vallois [27, 28]. On the other hand, W H is a Gaussian process, and (1.1) can be defined for deterministic processes a by way of an L2 isometry: see, for example, Nor- ros, Valkeila and Virtamo [21] or Pipiras and Taqqu [23]. With the help of stochastic calculus of variations (see [22]) this integral may be extended to random processes a. In this case, the stochastic integral (1.1) is a di- vergence operator, that is the adjoint of a stochastic gradient operator (see the pioneering paper of Decreusefond and Ustunel [9]). It must be noted that Duncan, Hu and Pasik-Duncan [12] have defined the stochastic integral in a similar way by using Wick product. Feyel and de La Pradelle [13], Ciesielski, Kerkyacharian and Roynette [4] also used the Gaussian property of W H to prove that W H belongs to suitable function spaces and construct a stochastic integral. 3 Eventually, Alos, Mazet and Nualart [1] have established, following the ideas introduced in a previous version of this paper, very sharp sufficient conditions that ensures existence of the stochastic integral (1.1). The construction of the stochastic integral. The starting point of our ap- proach is the following representation of fractional Brownian motion given by Decreusefond and Ustunel [9] t H H W (t) = Z K (t;s)dB(s); 0 where H 1 (t s) 2 1 1 1 t (1.2) KH(t;s) = − − F(H ; H;H + ;1 )(s < t); Γ 1 2 2 2 s (H + 2 ) − − − where F denotes Gauss hypergeometric function and (B(t);t 0) is a stan- dard Brownian motion. ≥ The first step we take is to define the integral of deterministic functions H H H (section 4). Observing that K (t;s) = It 1[0;t](s) where It is the integral operator t H H ∂ H (1.3) It f (s) = K (t;s) f (s) + Z ( f (u) f (s)) 1K (u;s)du; s − we define t t H H (1.4) Z f (s)dW (s) = Z It f (u)dBu 0 0 for suitable deterministic functions. In order to extend formula (1.4) to random processes f , we introduce the key idea of this paper, which is to approximate W H by processes of the type t (1.5) WK(t) = Z K(t;s)dB(s)(t [0;T]) 0 2 with a kernel K smooth enough to ensure that WK is a semi-martingale (Proposition 2.5). Then, for a such a semi-martingale kernel K and a good integrand a, we prove the following generalization of (1.4) t t t (1.6) Z a(s)dWK(s) = Z Ita(s)dB(s) + Z ItDsa(s)ds; 0 0 0 H where Dsa denotes the stochastic gradient and It is defined as It . In sections 2 and 3 topologies are defined on the space of kernels and the space of integrands. These are not ’natural’ topologies but ad hoc ones, whose sole aim is to fulfill the following requirements: 1. The mapping (a;K) ta(s)dW (s) is bilinear continuous (Proposi- R0 K tions 6.1). ! 2. Rough kernels such as KH are in the closure of the space of semi- martingale kernels. 4 P. CARMONA, L. COUTIN, AND G. MONTSENY 3. For nice integrands a the process t ta(s)dW (s) has a continuous R0 K modification (Theorem 7.1). ! Eventually, we are able to take limits in the Ito’sˆ formula established for n semi-martingale kernels in Proposition 8.1. Let Cb be the set of functions f whose derivatives, up to the order n N, are continuous and bounded. Privault proved that for f C2, 2 2 b t t H H H 1 H 2H 1 f (W (t)) = f (0)+Z f 0(W (s)):dW (s)+ Z f 00(W (s))(cH2Hs − )ds; 0 2 0 t H H 2 where 0 f 0(W (s)):dW (s) is the L -limit of divergences (Skorokhod in- R 3 tegrals). For H > 1=4 and f Cb, we show that this integral, which is not in general the limit of Riemann2 sums, is a Skorokhod integral with respect to the driving Brownian motion: t H H H (1.7) f (W (t)) = f (0) + Z It f 0(W )(s)dBs 0 t 1 H 2H 1 + Z f 00(W (s))(cH2Hs − )ds: 2 0 By the same procedure, we have been able to prove an Ito’sˆ formula for H > 1=6, in Proposition 8.11. We do not give here this formula since it is complex (5 lines) and does not seem to be an easy starting point for a generalization to every H (0;1). 2 To end this rather lengthy introduction we try to give an answer to a question that nearly everyone involved with stochastic integration with respect to Fractional Brownian motion has asked us. “What is the difference between the stochastic approach and the pathwise approach to integration with respect to Fractional Brownian motion?” H H On the one hand we prove in the Appendix that for W1 ;W2 two indepen- dent fractional Brownian motions of index 1=4 < H < 1=2, the integral t H H Z W1 (s)dW2 (s) 0 cannot be defined for the classical and generalized pathwise integrals.
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